Polarization singularities (PSs) have been under research for years, theoretically and experimentally [1-18]. In two-dimensions, PSs include C-points with half integral charges, where the light beam is circularly polarized, and L-lines, where the light beam is linearly polarized. In addition, there also exist V-points with integral indices [6]. V-points are generated from spatially varying linearly polarized fields while here we only focus on elliptical ones. Generally the winding number and the number of separatrices divide C-points into three kinds, Lemon, Monstar, and Star. Recently, high-order polarization singularities [19-21] and polarization singularity arrays [22-24] have also been proposed, which provide new directions for the study of polarization singularities.

The conventional generation of PSs is based on the superposition of two orthogonal circularly polarized components with optical vortex. For the optical vortex, it can be on-axis or off-axis. But this will not change the topological charge of the synthetic light field, but only changes the distribution location of the polarized ellipses and the position of the circularly polarized point. But there still exists another situation, that is a beam can carry more than one optical vortex [25-27].

In this article, we numerically generated hybrid PSs with the way to generate a Poincaré beam [28], which means superimposing two orthogonal circularly polarized components of a Gauss beam and a Laguerre-Gauss (LG) beam. But here we replace the LG beam with a Gauss beam with two optical vortices (OVs). The generated configuration is no longer Lemon or Star, but becomes the hybridization of them. The topological charge is no longer ±1/2, but the configuration is not high-order PSs neither. We take this as an intriguing discovery.

We set the origin to be at the focus of the two beams, and the beams propagate along the z axis. The Gauss beam at z = 0 can be expressed as:

where u₀ is a constant electric field amplitude, w₀ is the beam waist. As for the other component, we assume the topological charges of the two OVs are (+1, +1) or (+1, −1), and they are symmetrically distributed about the origin in the x-axis. So the Gauss beam with OVs in the initial plane can be expressed as [29-30]:

where r_k is a constant to define the location of the OVs.

As proposed in [24], the synthetic light field is generated with a circular basis as:

where u₁ and u₂ are the two components mentioned before, parameter γ regulates the intensity profile of the beam. ê_L and ê_R are the circular basis vectors, and in Cartesian coordinates, they are defined as:

Now let’s take an example. We set u₁ to be the Gauss beam with OVs of (+1,+1) and u₂ to be the initial Gauss beam. Substituting Eqs. (1) and (2) into Eq. (3), we can get:

After simple calculations [5], we can generate the normalized Stokes parameters as:

Now we will carry numerical simulation. For simplicity, we set u₀ =1, w₀ =1 mm, r_k = 0.5 mm, γ = π / 4, and the scale of the field to be 4 mm × 4 mm. Before we calculate the polarization configuration of the optical field, we firstly give the visualization of 2D and 3D intensity distribution of the Gauss beam with OVs of (+1, +1), shown as Fig. 1.

With the Stokes parameters of Eq. (6), we can draw the polarization ellipses distribution of the section, shown as Fig. 2(a). Here we didn’t show the polarization states of the whole light field, but only captured the center part of our concern as 2.4 mm × 2.4 mm. The polarization configurations analyzed below are the same scale except specific caption. The background is the intensity distribution of the synthetic field. The green ellipses represent the polarization states of the field. The yellow line, where the Stokes parameter S₃ = 0, connecting linearly polarized points is the boundary of opposite handedness. The light field is left-handed polarized out of the yellow loop while right-handed inside. The magenta lines in the figure are where S₁ = 0 and the blue lines are S₂ = 0, so intersection points of the magenta and blue lines are the PSs, represented as red circles in the figure. With the parameters S₁ and S₂, we can also generate the Stokes phase of the field, shown as Fig. 2(b). The colorbar on the right in Fig. 2 represents the corresponding value of the phases, which will not be illustrated in the figures later in the article.

Now we focus on the polarization states inside of the yellow line. If we take this area as a unit, the topological charge of the section is −1 because the azimuthal angles of the ellipses increase 2π in a clockwise sense. However we cannot take this configuration as a high-order PS. If we explore the polarization ellipse distributions around each PS, we find there exist two Stars inside of the area. That means there are two PSs inside of the area while there only exists one for high-order PSs. It looks like a combination of two Stars but it is not the “Star-Star” type in Vasnetsov’s article [31]. This is because in their configuration there exists a boundary between the two Stars, which means the handedness of the two PSs is opposite.

In Fig. 2(a), we can see that there does not exist any boundary between the two Stars, and they seem to “hybridize” together. In addition, there only exists one such a polarization configuration in the field, so we cannot take this as a polarization singularity array. So we consider this as an intriguing polarization singularity, and we name it as “twin-star”. Actually when a high-order V-point diffracts through triangular apertures, it will disintegrate into lower order C-points with such index [32]. The topological configuration will change into hybridization of two Lemons if we exchange u₁ and u₂, shown as Fig. 2(c), and now the topological charge changes to +1. In the figure, there doesn’t exist a boundary between the two Lemons either, so we may also take these two Stars as a unit, named as “twin-lemon”. Fig. 2(d) also illustrates its corresponding Stokes phase.

Similarly, if the topological charges of OVs of u₁ are (+1, −1), we can express the synthetic field as:

and now the normalized Stokes parameters turn into:

The polarization state of the field is shown as Fig. 3(a), and we found something interesting in the area. Without doubt in the horizontal direction there are two Lemons, represented by red circles. However, in the vertical direction, at the positions of S₁ = 0 and S₂ = 0 intersect, represented by two black dashed circles, there exist two Star-like topological configurations. The azimuthal angles of the ellipses around them increase clockwise by π , so the winding numbers of these locations are both −1/2, but clearly the ellipticity at these two points doesn’t equal to 1. As for the topological charge of the whole domain, we found that the azimuthal angles of the ellipses have not varied from 0 to 2π totally, so the winding number here, we consider its value to be 0. Similarly, we can interchange u₁ with u₂, the polarization states now are shown as Fig. 3(c). This time there are two Stars in the horizontal direction while there are two Lemon-like configurations in the vertical direction. We also think the topological charge of the area is 0.

Here we notice that the topological charge of the area has relation with the number and category of the PSs inside. In Fig. 2(a), the topological charge is −1, corresponding to the sum of the topological charges of two Stars, while the topological charge is +1 corresponding to two Lemons in Fig. 2(c). As for Fig. 3(a), the addition of topological charges of two Lemons and two Star-like configurations is 0, corresponding to the topological charge of the area. This conclusion also applies to Fig. 3(c).

We know that the topological configuration is determined by the amplitude and phase distribution of the two components, next we will discuss these two situations. To change the amplitude distribution of one of the components, we use an LG beam with OVs of (+1, +1) to replace the Gauss beam with OVs. We assume the location condition for the OVs remains unchanged, and we set the parameter p of the LG beam to be 0. So in the plane of z = 0 the LG beam can be expressed as:

We set the constant parameters same as the previous occasion, and also give the visualization of 2D and 3D intensity distribution of the LG beam as Fig. 4. Here we notice that in the center of the LG beam the intensity is 0, this is the main difference with the Gauss beam with OVs.

We set u₁ to be the LG beam with OVs and u₂ to be the initial Gauss beam, then we repeated the calculation procedure and generated the polarization states of the light field as Fig. 5(a). The topological configuration looks the same as Fig. 2(a), but we notice that there exists another circularly polarized point, represented as a black circle in Fig. 5(a), in the center of the area. This circularly polarized point results from the amplitude distribution of the two components. In the center of the area, the amplitude of the LG beam is 0, so the polarization here is totally decided by the Gauss beam. Obviously this circularly polarized point is not a PS. So strictly speaking, these two PSs are just Star-like configurations, but we still think the topological configuration of the area keeps invariable compared with Fig. 2(a) although the polarization states get changed inside. In another way, the Stokes phase illustrated by Fig. 5(b) is the same as in Fig. 2(b), that means the topological configuration of the two occasions should be the same, and the difference of the polarization states from Figs. 5(a) and 2(a) exactly ascribes to the amplitude difference of the fundamental Gaussian beam and the LG beam. By exchanging u₁ and u₂ we can get the polarization states as Fig. 5(c).

We know that the PSs come from the OVs of the components, and the topological charge of the OV will affect the kind of PSs. Here we will discuss the effect of another constant parameter, r_k. We set u₁ to be the Gauss beam with OVs of (+1, +1) and u₂ to be the initial Gauss beam. We set r_k to be two extreme cases, meaning r_k = 0 and r_k = 1 mm. The corresponding polarization states of the synthetic field are shown as Figs. 6(a) and 6(b) respectively. Here the scale of the captured field is 1.6 mm × 1.6 mm. In Fig. 6(a), the configuration turns into a high-order PS, a hyperstar with I_C = −1 [19]. While in Fig. 6(b), the hybrid “twin-star” gets separated into two Stars. But we need to notice there still doesn’t exist a boundary between the two Stars, so the handedness of the two Stars is the same, meaning this is also the “Star-Star” type in [31]. Compare these figures with Fig. 2(a) we found that r_k determines the location of the PSs and makes the polarization states get changed, but doesn’t account for the topological configuration of the light field until the critical situations as Figs. 6(a) and 6(b). Similarly we can generate a hyper-lemon and two separated Lemons by exchanging u₁ and u₁, shown as Figs. 6(c) and 6(d) respectively.

In the end, we further explore the effects brought by parameter γ . Obviously this parameter characterizes the amplitude ratio of the two components, and when γ takes 0 or π / 2, there only remains one component, and all of the field should be circularly polarized. We set u₁ to be the Gauss beam with OVs of (+1, +1) and u₂ to be the initial Gauss beam. In addition, we set γ to be π / 8, π / 4 (Fig. 2(a)), 3π / 8 and π / 2, and the corresponding polarization states of the fields are shown from Figs. 7(a) to 7(d), respectively. Here we exhibit the whole field to observe the variation. As the parameter γ increases, the background intensity pattern varies a lot until only one component is left (γ = π / 2), and the area of right-handed polarization becomes bigger. While inside the yellow loops, the topological configurations are similar. When γ increases to π / 2, the yellow loop no longer exists, and all of the field is right-handed polarized.

According to [25], we know that the propagation of OVs will lead to the rotation and separation of them. Therefore, the polarization singularities inside of the topological configuration will get rotated and separated, which means the configuration could be split into two polarization singularities after a certain propagation distance. It seems that such a configuration is not stable, but this property could be applied in particle manipulating. If we want to manipulate two specific particles relatively close together, we can use the synthetic light beam to rotate and separate them first.

In this article, we numerically simulated hybrid polarization singularity configurations based on the Gauss beam with two OVs. These polarization configurations have relations with the lowest-order PSs, but are different from the configurations proposed over these years. We also discussed the effects of the amplitude distribution, the separation distance of OVs, and the relative amplitude ratio parameter γ of the components to the polarization configurations, and found out in some critical situations the intriguing configurations turn into high-order PS or two lowest-order PSs. In addition, we figured out the relationship between the topological charge of the polarization configurations and the PSs inside. Our research may provide a theoretical basis for expanding the variety of the polarization singularity configuration.